A Coordinate Free Formulation of Effective Diffusion on Channels

TL;DR

This paper develops a coordinate-free mathematical framework using exterior calculus to derive an effective diffusion coefficient for diffusion processes in curved spaces, enhancing understanding beyond coordinate-dependent methods.

Contribution

It introduces a novel coordinate-free formula for the effective diffusion coefficient in complex geometries using exterior calculus techniques.

Findings

Derived a coordinate-free formula for effective diffusion coefficient D

Applicable to both infinite and finite transversal diffusion rates

Provides deeper geometric insight into diffusion processes

Abstract

We study diffusion processes in regions generated by sliding a cross section by the phase flow of vector filed on curved spaces of arbitrary dimension. We do this by studying the effective diffusion coefficient D that arises when trying to reduce the n-dimensional diffusion equation to a 1-dimensional diffusion equation by means of a projection method. We use the mathematical language of exterior calculus to derive a coordinate free formula for this coefficient in both infinite and finite transversal diffusion rate cases. The use of these techniques leads to a formula for D which provides a deeper understanding of effective diffusion than when using a coordinate dependent approach.